[1]} use additive concave welfare functions to generate Lorenz-efficient rankings. Let \(\phi (x, \alpha )=x^\alpha \)  if \(\alpha >0\) , \(\phi (x, \alpha )=\log (x)\)  if \(\alpha =0\)  and \(\phi (x, \alpha )=-x^\alpha \)  if \(\alpha <0\) . [1]} use concave welfare functions of the form:
\(g^{\rm user}({u}) = \sum _{i=1}^n \phi ({u}_i, \alpha _1)&&g^{\rm item}({v}) = \sum _{j=1}^m \phi ({v}_j, \alpha _2)\) 
